Generalized Stević-Sharma type operators from derivative Hardy spaces into Zygmund-type spaces

Let $u, v$ be two analytic functions on the open unit disk ${\mathbb D}$ in the complex plane, $\varphi$ an analytic self-map of ${\mathbb D}$, and $m, n$ nonnegative integers such that m < n . In this paper, we consider the generalized Stević-Sharma type operator $T_{u, v, \varphi}^{m, n}f(z) = u(z)f^{(m)}(\varphi(z))+v(z)f^{(n)}(\varphi(z))$ acting from the derivative Hardy spaces into Zygmund-type spaces, and investigate its boundedness, essential norm and compactness

Weighted composition operators and weighted conformal invariance

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de Lectura: 27-07-2021En esta tesis tratamos ciertos problemas relacionados con los operadores de composici´on ponderados. Estudiamos c´omo act´uan estos operadores en espacios de funciones anal´ıticas en D o en un dominio acotado Ω ⊂ C. En primer lugar nos centramos en una familia amplia de espacios de Hilbert de funciones anal´ıticas en el disco unidad, los cuales satisfacen solamente un n´umero reducido de axiomas y cuyo n´ ucleo reproductor tiene la forma usual. A estos espacios se les llama espacios de Hardy con peso. En estos espacios caracterizamos los operadores de composici´on ponderados que son co-isom´etricos (equivalentemente, unitarios). El resultado principal nos revela una dicotom´ıa al identificar una familia especifica de espacios de Hardy con peso como los ´unicos espacios en los cuales existen operadores no triviales de este tipo. La segunda parte de la tesis est´a dedicada a explorar una clase de espacios de funciones anal´ıticas los cuales comparten una cierta propiedad de invariancia conforme ponderada. Para ser m´as preciso, en esta parte presentamos una aproximaci´on general a los espacios que son invariantes bajo los operadores Wϕα, definidos por Wϕαf =(ϕ')α(f ◦ ϕ) con α> 0 y ϕ ∈ Aut(D). Podemos observar que muchos de los espacios de Banach de funciones anal´ıticas cl´asicos como los espacios de crecimiento de Korenblum, los espacios de Hardy, los espacios de Bergman con peso y ciertos espacios de Besov son invariantes bajo estos operadores. Entre otras cosas, en esta parte identificamos el espacio m´as grande, el m´as peque˜no y el “´unico” espacio de Hilbert que satisface esta propiedad de invariancia ponderada para un α> 0 dado. En la ´ultima parte consideramos espacios de Banach abstractos de funciones anal´ıticas en un dominio acotado general los cuales s´olo satisfacen unos pocos axiomas. A continuaci´on ponderados invertibles on, describimos todos los operados de composici´(equivalentemente, sobreyectivos) que act´uan sobre estos espaciosThis thesis treats a number of problems related to weighted composition operators. We study how these operators act on the spaces of analytic functions in D or in a bounded domain Ω ⊂ C. We first focus on a large family of Hilbert spaces of analytic functions in the unit disc which satisfy only a minimum number of axioms and whose reproducing kernels have the usual natural form. These spaces are called weighted Hardy spaces. In these spaces, we characterize the weighted composition operators which are co-isometric (equivalently, unitary). The main result reveals a dichotomy identifying a specific family of weighted Hardy spaces as the only ones that support non-trivial operators of this kind. The second part of the thesis is devoted to exploring a class of spaces of analytic functions which share certain weighted invariant property. More precisely, in this part we present a general approach to the spaces which are invariant under the operators Wϕα, defined by Wϕαf =(ϕ ')α(f ◦ ϕ) with α> 0 and ϕ ∈ Aut(D). We observe that many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces are invariant under these operators. Among other things, we identify the largest and the smallest as well as the “unique” Hilbert space satisfying this weighted invariant property for a given α> 0. In the last part, we consider abstract Banach spaces of analytic functions on general bounded domains that satisfy only a minimum number of axioms. Then, we describe all invertible (equivalently, surjective) weighted composition operators acting on such sp

Note on a new class of operators between some spaces of holomorphic functions

The boundedness and compactness of a new class of linear operators from the weighted Bergman space to the weighted-type spaces on the unit ball are characterized

Weighted composition operators from . . .

Let H B denote the space of all holomorphic functions on the unit ball B. Let u ∈ H B and ϕ be a holomorphic self-map of B. In this paper, we investigate the boundedness and compactness of the weighted composition operator uC ϕ from the general function space F p, q, s to the weightedtype space H ∞ μ in the unit ball

Thermodynamic Geometry: Evolution, Correlation and Phase Transition

Under the fluctuation of the electric charge and atomic mass, this paper considers the theory of the thin film depletion layer formation of an ensemble of finitely excited, non-empty $d/f$-orbital heavy materials, from the thermodynamic geometric perspective. At each state of the local adiabatic evolutions, we examine the nature of the thermodynamic parameters, \textit{viz.}, electric charge and mass, changing at each respective embeddings. The definition of the intrinsic Riemannian geometry and differential topology offers the properties of (i) local heat capacities, (ii) global stability criterion and (iv) global correlation length. Under the Gaussian fluctuations, such an intrinsic geometric consideration is anticipated to be useful in the statistical coating of the thin film layer of a desired quality-fine high cost material on a low cost durable coatant. From the perspective of the daily-life applications, the thermodynamic geometry is thus intrinsically self-consistent with the theory of the local and global economic optimizations. Following the above procedure, the quality of the thin layer depletion could self-consistently be examined to produce an economic, quality products at a desired economic value.Comment: 22 pages, 5 figures, Keywords: Thermodynamic Geometry, Metal Depletion, Nano-science, Thin Film Technology, Quality Economic Characterization; added 1 figure and 1 section (n.10), and edited bibliograph

Differentiation and composition on the Hardy and Bergman spaces

Banach spaces of analytic functions are defined by norming a collection of these functions defined on a set X. Among the most studied are the Hardy and Bergman spaces of analytic functions on the unit disc in the complex plane. This is likely due to the richness of these spaces. An analytic self-map of the unit disc induces a composition operator on these spaces in the natural way. Beginning with independent papers by E. Nordgren and J. V. Ryff in the 1960\u27s, much work has been done to relate the properties of the composition operator to the characteristics of the inducing map. Every composition operator induced by an analytic self-map of the unit disc is bounded on the Hardy and Bergman spaces. Differentiation is another linear operation which is natural on spaces of analytic functions. Unlike the composition operator, the differentiation operator is poorly behaved on the Hardy and Bergman spaces; that is, it is not a bounded operator. We define a linear operator, possibly unbounded, by applying composition followed by differentiation; that is, for f in a Hardy or Bergman space and an analytic self-map of the disk, $\phi$,DC\sb\phi(f)=(f\circ\phi)\prime.We have found a characterization for the boundedness of this operator on the Hardy space in terms of the inducing map. The operator is bounded exactly when the image of the self-map of the disc is contained in a compact subset of the disc. In contrast, we have found a self-map of the disc with supremum norm equal to one that induces a bounded operator on the Bergman spaces. In this setting we have found conditions necessary for boundedness, and conditions sufficient to imply boundedness. These conditions are closely related. The techniques used involve Carleson-type measures on the unit disc. A very general question arising out of this work involves relating boundedness of the differentiation operator to characteristics of these measures